A Modified Dai–Liao Conjugate Gradient Method Based on a Scalar Matrix Approximation of Hessian and Its Application

Abstract

We introduce and investigate proper accelerations of the Dai–Liao (DL) conjugate gradient (CG) family of iterations for solving large-scale unconstrained optimization problems. The improvements are based on appropriate modifications of the CG update parameter in DL conjugate gradient methods. The leading idea is to combine search directions in accelerated gradient descent methods, defined based on the Hessian approximation by an appropriate diagonal matrix in quasi-Newton methods, with search directions in DL-type CG methods. The global convergence of the modified Dai–Liao conjugate gradient method has been proved on the set of uniformly convex functions. The efficiency and robustness of the newly presented methods are confirmed in comparison with similar methods, analyzing numerical results concerning the CPU time, a number of function evaluations, and the number of iterative steps. The proposed method is successfully applied to deal with an optimization problem arising in 2D robotic motion control.

A Family of Multi-Step Subgradient Minimization Methods

“…However, (31) contradicts both (32) and (33). The resulting contradiction, taking into account (35) and (34), proves that the limit point can only be the minimum point.…”

“…We choose δ > 0, such that (29) will be satisfied. By virtue of conditions (34), there is K 0 , such that when j > K 0 , the relation will hold:…”

“…Dai and Liao [30] introduced CGM based on a modified conjugate gradient update parameter. Modifications of this method were later presented in [31][32][33][34].…”